Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2943 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(9,10,13,18,14,17,28,32)(11,19,21,30,22,33,26,15)(12,23,24,34,25,16,27,31)(20,35,36,29,37,40,38,39);(2,6)(4,7)(9,11,14,22)(10,15,17,30)(12,20,25,37)(13,26,28,21)(16,29,23,39)(18,33,32,19)(24,38,27,36)(31,35,34,40);(1,2)(3,4)(5,6)(7,8)(9,12,14,25)(10,16,17,23)(11,20,22,37)(13,27,28,24)(15,29,30,39)(18,34,32,31)(19,35,33,40)(21,38,26,36);(1,3,5,8)(2,7,6,4)(10,17)(15,30)(16,23)(18,32)(19,33)(29,39)(31,34)(35,40);(1,4,5,7)(2,3,6,8)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6524807);